Non-linear ODE solution

Dec 2015
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Earth
\(\displaystyle y''+2y=3e^x \).
 

romsek

Math Team
Sep 2015
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1,608
USA
$s^2 + 2 = 0\\
s = \pm i\sqrt{2}\\
y_h(x) = c_0 \cos(\sqrt{2}x) + c_1 \sin(\sqrt{2}x)
$

$y_p(x) = c_2 e^x\\
c_2e^x + 2c_2 e^x = 3e^x\\
c_2 = 1\\
y_p(x) = e^x\\
y(x) = y_h(x) + y_p(x) = c_0 \cos(\sqrt{2}x) + c_1 \sin(\sqrt{2}x) + e^x
$

This is a linear diff eq. It just has a driving function.